Schmidt number equation

The values of d and n are given in Table 3 typical values for can be found in Table 4. The exponent of 0.5 on the Schmidt number (l-L /PiLj) supports the penetration theory. Further examples of empirical correlations provide partial experimental confirmation of equation 78 (3,64—68). The correlation reflecting what is probably the most comprehensive experimental basis, the Monsanto Model, also falls in this category (68,69). It is based on 545 observations from 13 different sources and may be summarized as... 

The Einstein relation can be rearranged to the following equation for relating Schmidt numbers at two temperatures ... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

Another property of gases which appears in the Reynolds and the Schmidt numbers is the viscosity, which results from momentum transfer across the volume of the gas when drere is relative bulk motion between successive layers of gas, and the coefficient, y], is given according to the kinetic theoty by the equation... 

When bodr phases are producing eddies a more complicated equation due to Mayers (1962) gives the value of the mass transfer coefficient in terms of the Reynolds and Schmidt numbers which shows that die coefficient is proportional to... 

It is not possible to translate the above reasoning to turbulent flow, as turbulent flow equations are not reliable. However, in practice it is typical to assume that the same analogy is also valid for turbulent flow. Because of this hypothesis level, it is quite futile to use the diffusion factor D g in the Schmidt number instead we will directly use the number D g as in the Sherwood number. Hence in practical calculations Sc = v/D b-... 

Model experiments where free convection is the important part of the flow are expressed by the Grashof number instead of the Archimedes number, as in Eq. (12.61). The general conditions for scale-model experiments are the use of identical Grashof number, Gr, Prandtl number, Pr, and Schmidt number,, Sc, in the governing equations for the room and in the model. 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

For mass transfer, which is considered in more detail in Chapter 10, an analogous relation (equation 10.233) applies, with the Sherwood number replacing the Nusselt number and the Schmidt number replacing the Prandtl number. 

There has for long been uncertainty concerning the appropriate value to be used for the exponent of the Schmidt number in equation 10.225. SHERWOOD, PlGFORD and WlLKE 27 ... 

In defining a 7-factor (jd) for mass transfer there is therefore good experimental evidence for modifying the exponent of the Schmidt number in Gilliland and Sherwood s correlation (equation 10.225). Furthermore, there is no very strong case for maintaining the small differences in the exponent of Reynolds number. On this basis, the /-factor for mass transfer may be defined as follows ... 

Experimental results for fixed packed beds are very sensitive to the structure of the bed which may be strongly influenced by its method of formation. GUPTA and Thodos157 have studied both heat transfer and mass transfer in fixed beds and have shown that the results for both processes may be correlated by similar equations based on. / -factors (see Section 10.8.1). Re-arrangement of the terms in the mass transfer equation, permits the results for the Sherwood number (Sh1) to be expressed as a function of the Reynolds (Re,) and Schmidt numbers (Sc) ... 

For mass transfer to a surface, a similar relation to equation 12.117 can be derived for equimolecular counterdiffusion except that the Prandtl number is replaced by the Schmidt number. It follows that ... 

The above equations are applicable only when the Schmidt number Sc is very close to unity or where the velocity of flow is so high that the resistance of the laminar sub-layer is small. The resistance of the laminar sub-layer can be taken into account, however, for equimolecular counterdiffusion or for low concentration gradients by using equation 12.118. 

Schmidt number, multiparticle collision dynamics, real-time systems, 113-114 Schrodinger equation ... 

Hubbard and Lightfoot (HI la) earlier reported a Sc,/3 dependence on the basis of measurements in which the Schmidt number was varied over a very large range. The data did not exclude a lower Reynolds number exponent than 0.88, and reaffirmed the value of the classical Chilton-Colburn equation for practical purposes. Recent measurements on smooth transfer surfaces in turbulent channel flow by Dawson and Trass (D8) also firmly suggest a Sc13 dependence and no explicit dependence of k+ on the friction coefficient, with Sh thus depending on Re0,875. The extensive data of Landau... 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

The convective mass transfer coefficient hm can be obtained from correlations similar to those of heat transfer, i.e. Equation (1.12). The Nusselt number has the counterpart Sherwood number, Sh = hml/Di, and the counterpart of the Prandtl number is the Schmidt number, Sc = p/pD. Since Pr k Sc k 0.7 for combustion gases, the Lewis number, Le = Pr/Sc = k/pDcp is approximately 1, and it can be shown that hm = hc/cp. This is a convenient way to compute the mass transfer coefficient from heat transfer results. It comes from the Reynolds analogy, which shows the equivalence of heat transfer with its corresponding mass transfer configuration for Le = 1. Fire involves both simultaneous heat and mass transfer, and therefore these relationships are important to have a complete understanding of the subject. 

When the Schmidt number is greater than unity, addition of a scalar transport equation places a new requirement on the maximum wavenumber K. For Sc > 1, the smallest characteristic length scale of the scalar field is the Batchelor scale, 7b- Thus, the maximum wavenumber will scale with Reynolds and Schmidt number as... 

A comparison of equations 6.63 and 6.69 shows that similar forms of equations describe the processes of heat and mass transfer. The values of the coefficients are however different in the two cases, largely to the fact that the average value for the Prandtl number, Pr, in the heat transfer work was lower than the value of the Schmidt number, Sc, in the mass transfer tests. 

For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Cl see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30], The important difference... 

For packed beds, the use of these equations for predictions is limited by inaccuracy in the velocity-profile data. Therefore, Bischoff and Levenspiel (B14) used the equations in a semiempirical way for interpolating between the existing data. The results are shown in Figs. 15 and 16, for both empty tubes and packed beds. The heavy lines show the regions of experimental data, and the dashed lines, the interpolations. For sufficiently low flow rates, the curves lead into the reciprocal Schmidt number (modified by a tortuosity factor in packed beds). The data of Blackwell (B16, B17) at very low flow rates seems to verify this. At high flow rates, liquids and gases show no differences because of the... 

The form of Eq. (5-25) was suggested by noting that the first order curvature corrections to Eqs. (3-47) and (5-35) are near unity and by matching the expression to the creeping flow result, Eq. (3-49), at Re = E Equation (5-25) also represents the results of the application of the thin concentration boundary layer approach (Sc oo) through Eq. (3-46), using numerically calculated surface vorticities. Thus the Schmidt number dependence is reliable for any Sc > 0.25. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, using software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Rtf. To facilitate these calculations, the following data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sh, is defined as Sh = 0.04 Re0 75 Sc0-33, where Sc is the Schmidt number (2) osmotic pressure follows van t Hoff s equation, ie, 7r = iCRgTy where i is the number of ions (3)... 

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